Dose--response experiments are crucial in biomedical studies. There are usually multiple objectives in such experiments and among the goals is the estimation of several percentiles on the dose--response curve. Here we present the first non-parametric adaptive design approach to estimate several percentiles simultaneously via generalized Pólya urns. Theoretical properties of these designs are investigated and their performance is gaged by the locally compound optimal designs. As an example, we re-investigated a psychophysical experiment where one of the goals was to estimate the three quartiles. We show that these multiple-objective adaptive designs are more efficient than the original single-objective adaptive design targeting the median only. We also show that urn designs which target the optimal designs are slightly more efficient than those which target the desired percentiles directly. Guidelines are given as to when to use which type of design. Overall we are pleased with the efficiency results and hope compound adaptive designs proposed in this work or their variants may prove to be a viable non-parametric alternative in multiple-objective dose--response studies.